If the operation $ \oplus $ is defined by $ a \oplus b = a^{2} + b^{2} $ for all real numbers $ a $ and $ b $,then $ (2 \oplus 3) \oplus 4 = $

State whether the following statement is true or false. Justify: For an arbitrary binary operation $^*$ on a set $N$,$a \,^* \,a = a$ for all $a \in N$.

Which of the following is false?

Show that $-a$ is the inverse of $a$ for the addition operation '$+$' on $R$ and $\frac{1}{a}$ is the inverse of $a \neq 0$ for the multiplication operation '$\times$' on $R$.

Determine whether or not each of the definitions of $*$ given below gives a binary operation. In the event that $*$ is not a binary operation,give justification for this. On $Z^+$,define $*$ by $a * b = |a - b|$.

Let $*$ be a binary operation defined on the set of rational numbers $Q$. Determine whether the binary operation defined by $a * b = a^{2} + b^{2}$ for all $a, b \in Q$ is commutative.